in this study, by a non-negative homogeneous kernel k we prove some extensions of hardy's inequalityin two and three dimensions

We give conditions under which near-critical stochastic processes on the half-line have infinitely many or finitely cutpoints, generalizing existing results nearest-neighbour random walks to adapted with bounded increments satisfying appropriate conditional increment moments conditions. apply one of these deduce that a class transient zero-drift Markov chains in $\mathbb{R}^d$, $d \geq 2$, poss...